Given below are three classic triangles with which you should become familiar.

The six basic right-triangle ratios that you need to learn are:

Using the triangles and ratios given above, you should be able to state the EXACT value for each of the trig functions requested. All answers should be expressed as common fractions NOT decimals.

sine

cosine

tangent

cotangent

secant

cosecant

30º

37º

45º

53º

60º

Notice that three of these relationships are reciprocal functions:

Another group of functions are called co-functions. These functions quantify relationships between complementary angles.

Prior to the advent of scientific and graphing calculators, early trigonometry tables were usually set up based on these co-function relationships. Notice in the linked table how they display the fact that the co-functions for the complementary angles 37º and 53º are equal. The functions for the angles listed "down" the left side are across the top of the page while the functions for the angles listed "up" the right side are across the bottom of the page. For example, take a moment to see that the first column, "Sin," across the top is aligned with the column labeled "Cos" across the bottom.

On a unit circle, that is, a circle with radius equal to 1, the x-value of each co-ordinate represents the cosine of the central angle, the y-value of each co-ordinate represents the sine of the central angle.

Use the unit circle given above to determine the values for the sine and cosine of each quadrant angle given below. The remaining values for tangent, cotangent, secant, and cosecant can be calculated by using the functional relationships stated above. Once again, you should be able to state the EXACT value for each of the trig functions requested. All answers should be expressed as common fractions NOT decimals.

sine

cosine

tangent

cotangent

secant

cosecant

0º

90º

180º

270º

360º

A radian is a numerical ratio for any central angle that compares the magnitude of the intercepted arc length to the length of the circle's radius. This tells us that when the central angle θ equals 1 radian, the arc length s equals the radius. The expression s = rθ represents this relationship.

Note that when the arc length equals an entire circumference,

s = rθ 2πr = rθ θ = 2π radians = 360º

When measured in radians, as θ → 0, tan θ → sin θ which in turn approaches θ. This is called the small angle approximation and incurs an error of no greater than 0.1% for angles less than 5º. You can verify these relationships by examining the values for θ, sin θ, tan θ in Table 2.

All of these angle values are often represented graphically when we speak of circular functions. Trigonometrically, we generally use the variable x when expressing angles in terms of radians and θ when expressing them in terms of degrees.

Another useful set of trig identities are the called the Pythagorean Identities. These identities are based on line values drawn to a unit circle.

The double-angle formulas for sine and cosine.

sin 2θ = 2 sin θ cos θ cos 2θ = cos 2θ - sin 2θ

When solving for a missing side or angle is a triangle, there are two important relationships that apply to any triangle that can make your job easier: the Law of Sines and the Law of Cosines.

The Law of Sines, , can be used to solve for a missing side or angle in a general triangle when you know either

two sides and an angle opposite one side, or

only one side and all of the angles

The Law of Cosines can be used to solve for a missing side in a general triangle when you know the other two sides and their included angle.